K-theory of endomorphisms, Witt vectors, and cyclotomic spectra

Seminar/Forum

Of one the more interesting endofunctors of the category of categories is the one which associates to a category \(C\) its category of endomorphisms \(End(C)\). If \(C\) is a stable infinity category then \(End(C)\) is as well, and the associated K-theory spectrum \(KEnd(C):=K(End(C))\) is called the K-theory of endomorphisms of \(C\). Using calculations of Almkvist together with the theory of noncommutative motives of Blumberg-Gepner-Tabuada, we classify equivalence classes of endomorphisms of the \(KEnd\) functor in terms of a noncompeleted version of the Witt vectors of the polynomial ring \(\mathbf{Z}[t]\), answering a question posed by Almkvist in the 70s. As applications, we obtain various lifts of Witt rings to the sphere spectrum as well as a more structured version of the cyclotomic trace via cyclic K-theory, as studied in recent work of Kaledin and Nikolaus-Scholze.